Intertwining local (adjacency) metric dimension with the clique number of a graph

Abstract

Let $G$ be a simple connected graph with order $n(G)$, local metric dimension ${\dim }_l(G)$, local adjacency metric dimension ${\dim }_{A,l}(G)$, and clique number $\omega (G)$, where $G\ncong K_{n(G)}$ and $\omega (G)\ge 3$. It is proved that ${\dim }_{A,l}(G)\le \lfloor \left(\frac{\omega (G)-2}{\omega (G)-1}\right)n(G)\rfloor$. Consequently, the conjecture asserting that the latter expression is an upper bound for ${\dim }_l(G)$ is confirmed. It is important to note that there are infinitely many graphs that satisfy the equalities.